Next: Landau Collision Operator
Up: Collisions
Previous: Two-Body Coulomb Collisions
Rutherford Scattering Cross-Section
Consider a particle of type
, incident with relative velocity
onto an ensemble of particles of type
with number density
.
If
is the probability per unit time of the particle being scattered into the range of solid angle
to
, then the differential scattering cross-section,
, is defined via (Reif 1965)
![$\displaystyle p_1({\mit\Omega})\,d{\mit\Omega} = n_2\,u\,\frac{d\sigma}{d{\mit\Omega}}\,d{\mit\Omega}.$](img754.png) |
(3.83) |
Assuming that the scattering is azimuthally symmetric (i.e., symmetric in
), we can
write
. Now, the probability per unit time of a collision having an impact parameter in the range
to
is
![$\displaystyle p_1(b)\,db = n_2\,u\,2\pi\,b\,db.$](img758.png) |
(3.84) |
Furthermore, we can write
![$\displaystyle p_1({\mit\Omega})\,\left\vert\frac{d{\mit\Omega}}{db}\right\vert = p_1(b),$](img759.png) |
(3.85) |
provided that
and
are related according to the two-particle scattering law, Equation (3.82).
It follows that
![$\displaystyle \frac{d\sigma}{d{\mit\Omega}} = \frac{2\pi\,b}{\vert d{\mit\Omega}/db\vert}.$](img760.png) |
(3.86) |
Equation (3.82) yields
![$\displaystyle \frac{d{\mit\Omega}}{db} = 2\pi\,\sin\chi\,\frac{d\chi}{db} = -2\...
...\left(\frac{4\pi\,\epsilon_0\,\mu_{12}\,u^2}{e_1\,e_2}\right)2\,\sin^2(\chi/2).$](img761.png) |
(3.87) |
Finally, Equations (3.82), (3.86), and (3.87) can be combined to give the so-called
Rutherford scattering cross-section,
![$\displaystyle \frac{d\sigma}{d{\mit\Omega}} = \frac{1}{4}\left(\frac{e_1\,e_2}{4\pi\,\epsilon_0\,\mu_{12}\,u^2}\right)^2 \frac{1}{\sin^4(\chi/2)}.$](img762.png) |
(3.88) |
It is immediately apparent, from the previous formula, that two-particle Coulomb collisions are dominated by
small-angle (i.e., small
) scattering events.
Next: Landau Collision Operator
Up: Collisions
Previous: Two-Body Coulomb Collisions
Richard Fitzpatrick
2016-01-23